A four-dimensional model for the Ba–Ti–O dodecagonal quasicrystal

A four-dimensional structure model for the Ba–Ti–O dodecagonal quasicrystal is described.

The first oxide QC was reported recently by Fö rster et al. (2013) in a Ba-Ti-O ultra-thin film on a Pt(111) single-crystal substrate. The oxide QC was identified as a DDQC by observation of a 12-fold pattern in low-energy electron diffraction (LEED) images. In addition, the arrangement of protrusions observed in scanning tunneling microscopy (STM) images corresponds to dodecagonal Niizeki-Gä hler tiling (NGT), which is composed of three tiles, i.e. a triangle, square and 30 rhombus (hereafter rhombus) (Niizeki & Mitani, 1987;Gä hler, 1988), with an edge length of 6.85 Å . Atomic positions determined based on the STM images were statically analyzed and compared with the NGT by Schenk et al. (2019b). A second DDQC was more recently observed in an Sr-Ti-O ultra-thin film on a Pt(111) substrate (Schenk et al., 2017). The oxide DDQCs form on a periodic threefold substrate; therefore, the formation and propagation mechanism of the quasiperiodic long-range order are of significant interest. However, knowledge of the atomic structure of the DDQCs is crucial to understand the mechanism.
A tile decoration model of the Ba-Ti-O DDQC was proposed and investigated in detail by Cockayne et al. (2016). This model consists of three decorated tiles in the NGT, which leads to a stoichiometry of Ba 0.37 TiO 1.55 . The stability of the atomic structure was also investigated with hypothetical approximants (APs) to the DDQC using density functional theory (DFT) calculations, where the DFT-relaxed structures retained the ideal tile geometries. In addition, a simulated STM image based on the relaxed atomic structure of a hypothetical AP reproduced the experimental image, which indicates that the protrusions observed in the STM images are Ba atoms.
In higher-dimensional descriptions of DDQCs, the quasiperiodic atomic structure can be obtained as a three-dimensional (3D) section of a five-dimensional (5D) periodic structure that consists of so-called occupation domains (ODs). The OD has a two-dimensional (2D) shape defined in the 2D complementary space called the perpendicular space E ? , which is perpendicular to the 3D real space, called the parallel space, E k [see, for example, Yamamoto (1996), Janssen et al. (2007) and Steurer & Deloudi (2009)]. The 5D structure has two lattice constants, a d and c, and the atomic structure has a period along the c axis. The first 5D model was derived for a stable Ta-Te DDQC, which consists of fractal ODs (Yamamoto, 2004). The Ba-Ti-O DDQC was formed with monolayer thickness (Zollner et al., 2020a); therefore, we regard its atomic structure as a 2D structure. In such a case, the atomic structure is obtained as a 2D section of a four-dimensional (4D) structure in a 4D subspace orthogonal to the c axis. The higher-dimensional description allows atomic coordinates in the QC structure to be described with finite structural parameters, including 4D coordinates of the ODs and atomic displacement parameters in E k and E ? ; therefore, the construction of the 4D model is crucial to analyze the atomic structure of DDQCs. Here, we derive a 4D model for the Ba-Ti-O DDQC based on the tile decoration model of Cockayne et al. (2016).

A simple layer of Niizeki-Gähler tiling
In this section, we briefly describe a simple 4D model which places atoms at face-centre positions of the NGT (Gä hler, 1988). The coordinate system used in this study is based on that described in the literature (Yamamoto, 1996). The unit vectors of the 4D dodecagonal lattice d i (i = 1,2,3,4) are written using unit vectors in 2D E k , a 1 , a 2 , and 2D E ? , a 4 , a 5 , as with s 4 c 20 s 20 where a d is the lattice constant of the 4D dodecagonal lattice, and c n and s n are cosðn=6Þ and sinðn=6Þ, respectively. The projection of the d i onto E k and E ? is shown in Fig. 1. A 4D positional vector x = (x,y,z,u) is represented by using the d i , and the perpendicular-space component of the x is represented by subscript ? (the parallel-space component is represented by the subscript k). The coordinates in E k and E ? are given byQ Q k X andQ Q ? X, respectively, whereQ Q k andQ Q ? are the upper and lower 2 Â 4 part of the transposed matrix of Q in equation (2), respectively, and X is a transposed matrix of (x,y,z,u). Hereafter, we consider the 4D model with a lattice constant of the 4D dodecagonal lattice a d equal to 8.39 Å . This lattice constant corresponds to the edge length of the tile elements given by 2a d /6 1/2 , which is equal to 6.85 Å and corresponds to the observed edge length in the STM images of the Ba-Ti-O DQC (Fö rster et al., 2013).
The vertex positions of the NGT are obtained from an OD that is denoted as C a in the literature (Gä hler, 1988). The shape of C a is shown in Fig. 1(b). C a is situated at the vertex position of the 4D dodecagonal lattice with site symmetry 12mm, provided that a plane group of the resulting tiling of p12mm is assumed. The asymmetric part of C a is defined as a triangle where the vertices are represented by 4D positional vectors (0, 0, 0, 0), (À3 1/2 , 3 1/2 , 0, 0) ? /3 and (0, 1, 0, 0) ? in a unit of 2a d /6 1/2 . Fig. 2 shows a simple representation of the NGT at the vertex, edge-centre and face-centre positions. The OD in Fig. 2(a) generates the vertices of the tiling, which corresponds to C a . C a is subdivided into four parts which are assigned to a1, a2, a3 and a4, so that they distinguish four local configurations present in the tiling, as shown in Fig. 2(f). They distinguish the vertex positions as following; (1) a vertex position derived by OD a1 shares five triangles and two rhombuses; (2) a vertex position by OD a2 shares four triangles, one square and one rhombus; (3) a vertex position by OD a3 shares three triangles and two squares; and (4) a vertex position by OD a4 shares two triangles, one square and one rhombus. The areas of the asymmetric part of ODs a1-4 are [9 À 5(3) 1/2 ]/36, [7(3) 1/2 À 12]/18, [21 À 11(3) 1/2 ]/36 and [2(3) 1/2 À 3]/18; therefore, the frequencies of each local environment are determined as approximately 9.81, 7.18, 56.22 and 26.79%, respectively. Here, the area of each OD is divided by a d 2 . The ODs in Figs. 2(b)-2(e) generate the edge-centre and face-centre of each triangle, the face-centre of each square and the face-centre of each rhombus, respectively. The OD in Fig. 2(c) is subdivided into three parts, which are assigned to c1, c2 and c3 so that they distinguish three local configurations research papers present in the tiling, as shown in Fig. 2(g). They distinguish the triangles as following; (1) a triangle with a face-centre position derived by OD c1 shares no edge with other triangles; (2) a triangle with a face-centre position derived by OD c2 shares one edge with another triangle; and (3) a triangle with a facecentre position derived by OD c3 shares two edges with two other triangles. The areas of the asymmetric part of ODs c1, c2 and c3 are [9 À 5(3) 1/2 ]/36, [2(3) 1/2 À 3]/6 and [9 À 5(3) 1/2 ]/36; therefore, the frequencies of each triangle are determined as approximately 9.81, 80.38 and 9.81%, respectively. Here, the area of each OD is divided by a d 2 . The positions derived from the ODs in Figs. 2(a)-(d) are represented by the same colour in Fig. 2(h), and the Wyckoff positions, site symmetry, coordinates and ODs are summarized in Table 1, based on the Wyckoff positions of 5D dodecagonal space groups (Yamamoto, 2021). The 4D positional vectors that define the asymmetric part of ODs a-e are listed in Table 2. The ODs a-e can be derived from their asymmetric parts by applying the symmetry operations of their respective site-symmetry group.

4D model of Ba-Ti-O DDQC
To derive a 4D model of the Ba-Ti-O DDQC, we take into account the tile model proposed by Cockayne et al. (2016), as shown in Fig. 3(a). The model has the following features. First, the Ba atoms are situated at each vertex position. Second, the Ti atoms are located at the face-centres of each triangle, four positions in each square and two positions in each rhombus. Third, the O atoms are located at four positions in each square and two positions in each rhombus. The O atoms are located at three positions of each triangle; however, the position is dependent on the local environment of the triangle. When two triangles are neighbouring and sharing an edge, the O-atom positions near the edge in each triangle merge into one at a position close to the sharing edge (Cockayne et al., 2016).
Therefore, these positions should be half-occupied by O with an occupation probability of 1/2. When the occupational probability of the other sites is 1, the 4D model leads to a Ba:Ti:O ratio of 2(3) 1/2 :6 + 2(3) 1/2 :6 + 5(3) 1/2 , which is in good agreement with the stoichiometry of Ba 0.37 TiO 1.55 (Cockayne et al., 2016). The atomic arrangement of the Ba-Ti-O DDQC obtained from the 4D model is presented in Fig. 3(b), in which the atomic arrangement was generated with the following parameters: x 1 = u 1 = 1/4, x 2 = u 2 = 1/8, y 3 = z 3 = 3/8, y 4 = z 4 = 1/ 4 and y 5 = y 6 /2 = y 7 = u 5 /2 = u 6 = u 7 = 1/8. Point density of the QC is calculated from the ODs and the unit-cell volume of the higher-dimensional structure. In the 4D model of the DDQCs, the unit-cell volume is given by det|Q ij |, where Q ij is the 4 Â 4 matrix in equation (2). The point density is then given by = V ? /det|Q ij |, where V ? is the sum of the area of the ODs (Yamamoto, 1996). Because the point density of Ba in the 4D model equals 3 1/2 /a d 2 , the number densities for Ba, Ti and O are approximately 2.46 Â 10 À2 , 6.72 Â 10 À2 and 10.4 Â 10 À2 Å À2 , respectively. According to the STM observation, the areal density of the protrusions for the Ba-Ti-O DDQC is 3.2 Â 10 À2 Å À2 (Yuhara et al., 2020). This density is rather close to the point density of the Ba in the 4D model.   4D positional vectors x that define the asymmetric part of the ODs in Fig. 2, with the vectors presented in units of 2a d /6 1/2 .

Approximants
APs are important crystals for understanding the atomic structure of the QCs (Elser & Henley, 1985). The atomic structure of an AP is derived from a higher-dimensional structure of a QC by the introduction of an appropriate linear phason strain, and the AP exhibits a local structure similar to the QC [see, for example, Yamamoto (1996), Quiquandon et al. (1999), and references therein]. The formation of several long-range ordered structures with large unit cells has been found to date in Ba-Ti-O ultra-thin films (Fö rster et al., 2012). Two types of APs in the (Ba,Sr)-Ti-O ultra-thin films were reported recently, and these consist of three of the NGT tile elements, i.e. square, triangle and rhombus. The first is the sigma-phase approximant, which corresponds to an Archimedean tiling (3 2 .4.3.4) composed of squares and triangles, and it was identified in a Ba-Ti-O ultra-thin film on Pt(111) and Ru(0001) Fö rster et al., 2016;Zollner et al., 2020b). The second has a complex structure composed of squares, triangles and rhombuses with a large unit cell (a = 25.1, b = 37.7 Å and = 95.1 ), and it was identified in an Sr-Ti-O ultra-thin film (Schenk et al., 2017). Here, we present two of the simplest APs derived from the 4D model by the introduction of linear phason strain.   Table 3 Occupation domain (OD), atom, coordinates and occupancy (occ.) of the 4D model structure.
OD Atom Coordinates Occ.

Summary
We have derived the 4D model for the Ba-Ti-O DDQC by considering a tile model of the DDQC. 4D coordinates of the asymmetric part of the ODs were also provided in the 4D model. The atomic structure of the Ba-Ti-O DDQC results in a Ba:Ti:O ratio of 2(3) 1/2 :6 + 2(3) 1/2 :6 + 5(3) 1/2 , which corresponds to a stoichiometry of Ba 0.37 TiO 1.55 . In addition, the point density of Ba atoms in the 4D model was shown to be 3 1/2 /a d 2 % 2.46 Â 10 À2 Å À2 , which is in good agreement with the observed areal density of the protrusions on the STM images.
Lastly, we have shown that the atomic arrangement of the sigma-phase approximant and the hypothetical AP are derived from the 4D model by the introduction of a linear phason strain, and these structures consist of the same tile arrangement as in the Ba-Ti-O DDQC model.